Even under the assumption that \(A\)  is open, an h-principle for \(A\) -directed embeddings is not generally true. For example if \(L = S^2\)  and \(Y=\mathbb {R}^3\) , the h-principle for \(A\) -directed embeddings fails for any proper subset \(A \subseteq Gr_{3, 2}\) . In  [1]}, it is shown that an h-principle holds for all open \(A\) , if \(L\)  is an open manifold. Furthermore, the concept of convex integration is used there to prove an h-principle holds for \(A\) -directed embeddings of closed manifolds, under the assumption \(A\)  is open and ample. Rather than stating the original definition, we give the ampleness criterion 19.1.1 from  [2]}.
